A "change of basis" is an action performed in linear algebra, whereby a change in fundamental structure yields an entirely new viewpoint. This blog began as a record of a pedagogical change of basis for me, and continues as an ongoing account of my thoughts as I design and direct courses in mathematics at the University of North Carolina, Asheville.

Monday, August 31, 2009

Below is the most marvelous dialogue ever written by a student in response to one of my 280 homework questions. This particular question challenged the students to construct a dialogue in which the interlocutors got at the ideas underpinning quantifiers: what exactly is the difference between universal and existential quantifiers? The ensuing dialogue is thorough, clear, and exceedingly well-composed. It hits on just about every single point, subtle or not, you can expect a first-year prover to pick up on. Put simply, it's gorgeous.

Enjoy!

***

“I just don’t get it, any of it. It’s like, if universal means all, then shouldn’t the opposite one mean nothing? It doesn’t make sense that the opposite of ‘everything is this way’ is ‘something else exists.’ It feels like the opposite should be ‘nothing is this way.’”

“I can see where you’re coming from, and it seems like a lot of the confusion comes from thinking in terms of opposites rather than negation. In math, you don’t really need the opposite of a statement to disprove it, you just need the negation. Like, if I were to make a statement with a universal quantifier and say ‘all frat boys are jerks’—"

“That’s not true! David’s pretty nice.”

“Exactly! You just constructed a negation to my statement without even thinking about it. To prove that I was wrong in saying that all frat boys are jerks, you didn’t say ‘no frat boys are jerks,’ you just said that there exists a frat boy who is not a jerk.”

“Well, David’s not the only one. There are plenty of decent guys in fraternities. So it’s not an existential quantifier, because it’s more than one.”

“It’s still an existential quantifier if you’re talking about some instead of one. We use existential quantifiers when we want to talk about at least one, sometimes more than one, but not necessarily everything in the set we’re dealing with. I could say ‘there exists an integer that is not two’ and that would be an existential statement.”

“But it’s not true. There are a lot of integers that aren’t two, not just an integer that is not two. Actually, most integers are not two. Almost all of them are not two.”

“That’s true, but the point is that at least one integer, but not necessarily all integers, is not two. As long as the statement I’m making isn’t claiming anything about the entire set, it’s existential instead of universal.”

“So would the negation of that statement be ‘there is an integer that is equal to two’?”

“Not quite. To negate an existential quantifier, another existential quantifier won’t do. There’s plenty of room in the integers for things that are and aren’t two, so you can’t disprove my original assertion by saying that there is an integer that is two. You would need to say that in the set of all integers, there is no two.”

“That’s false. But I guess it’s supposed to be false, because what you said about an integer being not two was true, and to try to disprove it would be false.”

“Yes, the negation of a statement is always the opposite truth value of the original statement.”

“But why do I have to make a statement about everything in a set to disprove a statement about one thing in the set? Can’t I just say that the one thing doesn’t have the property that the original statement claimed it did? Before, you said I didn’t have to prove that every frat boy was a decent guy, just that there was one who was. Why is that wrong now?”

“To say something about the one thing in the set not having those properties, you kind of need to say something about everything in the set. Let’s try another non-math example. If I were to tell you that The Hop sells beef-flavored ice cream, how would you respond?”

“Eew. No it doesn’t.”

“What you said is the negation of my statement, and it uses a universal quantifier.”

“Didn’t sound like one.”

“No, it didn’t, but it will with some re-phrasing. To tell me that The Hop doesn’t sell beef-flavored ice cream, you are saying that none of the ice cream that they do sell is beef-flavored. So what you’re saying is that in the set of all the flavors sold at The Hop, every flavor shares the quality ‘not beef.’”

“That’s right! In this case, trying to use an existential quantifier would be like saying ‘they sell a lot of flavors that aren’t beef,’ and that still leaves room for the possibility that they also sell beef. But they don’t, because all of the flavors are not beef.”

Friday, August 28, 2009

We're just about done with the second week of classes, and things are going great. All of my classes are engaged this semester (I had perfect attendance, 66 out of 66 between the two sections, for the first quiz in Calc I), and there's profound and evident eagerness to learn on the part of many, if not most, of my students.

I hope it doesn't wear off!

Further bulletins as events warrant, but here are the highlights:

I've already had great conversations with 280 students about the effectiveness of dialogues in conveying deep mathematical concepts,

the very first homework committee volunteers, in their presentation to class, mentioned (without prompting from me) "audience" as a major consideration in constructing a solid solution to a math problem,

several students are already all-upons regarding Super Saturday,

I've passed out almost a dozen of the newly-printed math major booklets,

several of the Calc I students have already read up on the Newton v. Leibniz project on-line, even though I've not yet handed out the project description,

dozens of the Calc I students are reporting that they're doing rough initial drafts of their homework before crafting a clean final version, and

at least two 280 students have already begun TeXing the entirety of their homework.

Tuesday, August 25, 2009

It was still dark. Like most parents, mine were eager to spend the better part of their Saturday mornings in bed, except on the mornings when my dad had a yen to hit the trailhead early. At six years old, I was incapable of sleeping in past seven.

It was 1981. Space Ghost took turns with the standard stable of Warner Bros. characters on the giant and brilliantly-lit Curtis Mathis console television that dominated our tiny living room. The volume was turned down very, very low so that the sound wouldn't wake my mother, snoring away on the couch in the next room.

My attention wasn't on the TV screen, but rather on the toy I held clumsily in my hands, the Texas Instruments Little Professor, several ounces of hard yellow plastic skin surrounding high-tech electronic guts. Made to help children drill themselves on arithmetic, the toy spat out problems involving addition, subtraction, multiplication, and division. By then I'd learned the first three operations and begun to master them, but was still befuddled by the fourth.

On that morning I drilled myself with multiplication for a while (I'd only just learned that one), working problem after problem as the Little Professor added points and upped the difficulty of the problems it gave me. I missed a few here and there, but it wasn't long before I bored of that operation and decided to move on to something more challenging.

I'd not yet figured out division. Addition and subtraction were old hat, and multiplication I'd learned by conceiving of it as repeated addition, the only means I then had of computing products of multidigit numbers, being at that time unfamiliar with the formal method of "long" multiplication. But division? Fuhgeddaboutit.

That morning I had a hunch I wanted to follow up on: I suspected that just as subtraction served as an inverse operation to addition, "undoing" what addition did, division must serve as an inverse to multiplication. Thus, for example, if 6 x 8 were 48, a request to divide 48 by 6 would be answered by finding the number of sets of size 6 it would take to comprise all of 48 when unioned together. (I'm sure that I wasn't able to so clearly and succinctly summarize the process at the time, but my recollection of those general thoughts is quite clear, even now, nearly three decades later.)

Simple enough in principle, the details of this inverse operation were hard for me to carry out as soon as the superset had more than a few elements in it. Therefore when I switched the Little Professor over to division mode, the only questions I was able to answer unfailingly at first were those in which the divisor was 1. Nevertheless, my success with these simple problems lent credence to my theory regarding division's nature.

After a good deal of trial and error and after even more practice in the quick computation of products that it took me to "unwind" the multiplication once more to obtain the quotients I sought, I began to get better. Soon I was able to tackle the problems in which 2 appeared as the divisor, counting the number of 2s it took to make up the dividend I was given. Soon after that, 3s posed no problem, and I was on to 4s, 5s, and beyond.

My method was clumsy: it would be a long time before I learned long division, and until that time I'd have to resort to the protracted multiplication through repeated addition, done backwards, in order to solve even reasonably complicated division problems.

The revelation was too exciting to keep from my mother, and I ran into the room next door to wake her and show off what I'd done. Obviously I can't recall her exact words, but I'm sure they were something along the lines of "that's nice, kid. Show me in a few hours, when normal people are up."

I'd hardly be human had I not felt a jolt of euphoria on making the discovery I'd made for myself, for I believe that much of what makes us human is our desire to seek order and understanding of the world around us. There's no high on Earth like the one that making such a discovery gives.

Friday, August 21, 2009

Already I've noticed several signs that this semester's 280 students are a sharp lot, and with a little poking and prodding I'm sure they'll be capable of great things this semester.

For instance, they've already picked up on the subtleties of the proof and disproof of universal and existential statements, which subtleties might make fine fodder for a paragraph or two in the "textbook" chapter they'll be writing on mathematical statements. Namely, they recognized almost immediately, and on their own, that one can disprove a universal statement through counterexample, but not an existential one, paving the way to an easy understanding of negation, the topic for this coming Monday.

I like to think that their effortless mastery of these concepts comes as much from their preparation as from their perspicacity: it certainly helped to make sure all (or at least most) of them had read through the class's worksheet before class had been convened. I'm thinking it's going to pay off handsomely to designate discussion leaders as I've begun to do, also. Today the three discussion leaders provided the majority of the examples on the board, and did a fine job of explaining the choices they made.

This course's structure is in continual flux, but I think I'm more and more closely approximating an ideal design. My only worry at this point is that the "textbook" component of the course, though I'm quite sure it'll work pretty well this semester, would prove too complicated a project to put together in a larger class.

We'll see. I've admitted to the students that I'm not even sure how that aspect of the course will come together, and I'm as curious as they are (probably far more so) to see what comes of it.

Wednesday, August 19, 2009

Today was the second day of actual classes for me, and they went very well, substantially more smoothly than Monday's classes overall.

My 280 students did a remarkably good job in poring over the "Does good writing matter?" handout I've now used four times to encourage students to generate criteria for strong writing in nonmathematical subjects: they made a few observations no previous students have made, and they weren't the least bit shy in making them. For instance, they pointed out that the response I'd clearly intended to be the strongest one suffered from overuse of jargon and generalities, which can both be used to disguise ignorance; the "balance of power" buzzphrase was the one at which one group's ears perked. "People tend to write like that when they don't really understand what they're talking about," one student said. Nevertheless, most agreed that the second passage was still superior to the first, which despite its attention to detail frequently rambled far off topic. (Of course, the third, intentionally written to be sparse and grammatically disjointed, was everyone's pick for last place.)

Yes, the conversation regarding that handout was a lively one, and the ensuing one on the mathematical counterpart, "Does good mathematical writing matter?," was equally spirited. Already they're picking up on some important aspects of well-written proofs (complete and grammatically correct sentences, utmost clarity, etc.; as Belinda asked at one point, "so we should be writing as though we're writing to aliens who have no understanding of what we're talking about, trying to make it as clear as possible?"). When the time came for me to solicit volunteers for discussion leaders for the first "official" handout, several were more than willing.

So far the students seem sharp, outgoing, and eager to work.

Both sections of Calc I went well, too, as the students took turns presenting (in pairs) on the precalculus topics they'd been asked to review for today. Some of the presentations were understandably a little rough, but all were satisfactory, and some were positively outstanding. In the first section, those presentations dealing with the Vertical Line Test, asymptotes, and rational functions were particularly strong, involving appropriate and explicit examples. Ino and Nadia, for instance, were spot on in their discussion of asymptotes in the first section.

Today also brought me the chance to meet several more students during their "meet 'n' greet" interviews. A few were shy, as many were eager to set on the semester. It turns out that several (5 out of 15) of my 280 students are math licensure students, a disproportionately large number, as I mentioned a few posts back.

Tomorrow will be the third day of classes for Calc I, and a break from 280. Further bulletins as events warrant.

Oh, and: I've just put together a rough draft of my proposal for the 10th International Writing Across the Curriculum Conference in Bloomington, Indiana in May 2010. I hope to speak on the intentional disciplinary writing instruction I've been doing for the past two summers during the REU.

Tuesday, August 18, 2009

What a difference a day makes! Today's languor lies in stark contrast to the tumult that was yesterday. I've had a chance to get caught up on several little nagging tasks in during the free hours in between my "get to know ya" meetings with MATH 280 students. (I've got another meeting in just a few minutes.)

So far they're all professing eager anticipation for what's to come in the course. There's a little trepidation about LaTeX, but that's only natural. One student expressed slight annoyance at having to do a good deal of work in groups, indicating that he's the sort of person who prefers to do everything himself if he knows he's the one who's apt to do it best. (I understand the feeling: I was like that in college, and every now and then I still slip back into that mode.) Thankfully, though, he recognized the importance of collaborative work and seems completely willing to give it a go in our class.

Another benefit of having such a small class is the brevity of the course's "mixing time": suppose, for simplicity, that every group project will involve 3 students. If groups are assigned completely at random each time a group is convened for a group project, in a class of 15 students after a person has served on 5 groups, she or he can expect to have worked with roughly 51% of the people in the class. In a class of 30 students, 5 rounds of service only puts the one student in touch with roughly 29% of her or his classmates, on average. (Yes, I'm enough of a nerd that I just wrote some Mathematica code to generate the desired expected values for arbitrary class size and group size.) Put simply, a student more quickly comes into contact with a greater proportion of her or his class in a class of 15 people.

Not shocking, but it's nice to know that the numbers back it up!

Okay, that's all for today. I'm homeward bound quite soon, but I'm excited to find out what tomorrow holds for me, class-wise.

Monday, August 17, 2009

No time like the first day of class for the campus internet system to go farfufket. Connectivity to off-campus sites has been spotty since noon, and it's wreaked havoc with class activities, e-mail, and class prep as I'm trying to put out the fires set during the first section of Calc I this morning.

So what happened in that today that got my ever-lovin' panties in a big ol' knot?

My colleague who directs our Math Lab came in to give his Math Lab spiel, and to give the students pretty explicit instructions regarding the precalculus review "pretests" we've asked them to complete for placement purposes. These instructions, detailed and demonstrative as they were, took a good deal of time, and by the time I finished going over the syllabus (in what I felt were fairly broad strokes), there were precisely three minutes of class time left.

I didn't get a chance to solicit contact/background info from the students, and I didn't get a chance to ask them to sign up for a "meet 'n' greet" interview with me sometime in the first couple of weeks. This may not seem like that much of a big deal, but those of you who know my teaching style can vouch for me when I say that it's crucial to me to establish a good rapport and simpatico (for lack of a better term) with my students, right out of the gate. Both teaching and learning are optimally done in an environment of familiarity, support, comfort, and mutual respect, and such an environment is built by early and frequent efforts on the part of both instructor and students to meet one another on common ground.

Moreover, we had no time for review, as it took all of the measly amount of time we had left simply to explain the homework for Wednesday. (Doing this was of paramount importance, as they simply have to be prepared for Wednesday's class if we're going to get anything out of it.)

The second section of Calc I ran much more smoothly: the Math Lab/pretest spiel was dramatically abridged due to the aforementioned network outage (silver lining, anyone?), leaving more time to answer student questions and then proceed more leisurely to the homework for Wednesday.

Nevertheless, though I'm much more satisfied with my experience in the second section today, it was still far from ideal.

If I had to rate the first day, I'd give myself a B for 280, a C for Calc I, Section 1, and an A- for Calc I, Section 3. Combine that with an A for damage control and coping with chaos, and I might just make a B so far.

It'll all be calmer on Wednesday, folks, believe you me!

P.S.: I've already had a few very pleasant out-of-class exchanges with new students, and I've no doubt they'll be joined by myriad more as the semester goes on. Keep at it, my friends, keep at it!

My second section of Calc I went faaaaaaaaaaaar more smoothly than my first section. (Hooray for not having to deal with Educo at all!) Although it would have been nice to have had a bit more time (five minutes would have been really nice), everything went more or less according to plan, and already I feel much more comfortable with that section than I do with the first.

If any of my new students have followed up on my hints and have come to this blog for the first time, welcome! I hope you'll check in again every now and then to see what I'm thinking as we go forth into this new semester, and I hope you'll feel free to offer me feedback on the activities we design together in the classroom.

For now, get ready for Wednesday, try to get into some sort of groove, and have fun!

I think it went all right. The one drawback to the "complexity" of the course (HW committees, discussion leaders, now a student-authored "textbook") is that it now takes about 25 minutes to give a reasonable explanation of the course's components, and that's without going in to much detail.

14 people attended today, including two who aren't yet registered, so signs still point to a small class, roughly half the size of last semester's 280. I'm glad for that: it's conducive to a more intimate and supportive classroom dynamic. So far there are a couple of people who seem to be a bit more outgoing, inquisitive, and bold. If I can get them all to rally around a few natural leaders, it'll be a great team.

One point of note: most of the math majors in the class are in the Licensure concentration. Interesting.

Now I'm off to get ready for my first section of Calc I, and so must for the moment say adieu...

Though I've been eagerly anticipating today's arrival for the past few weeks, the first wave of true excitement didn't hit the shore until I crested the hill on the way into campus this morning. The sight of students and colleagues alike converging on the campus from all directions reawakened my enthusiasm.

Monday, August 10, 2009

As the first day of class in the Fall 2009 semester (too quickly) approaches, and as this blog passes its third birthday and nears its 300th post, I wanted to take a moment to find out who's out there.

I began writing this blog a little over three years ago more for myself than for anyone else. I intended it to serve as a means of organizing my thoughts on my teaching, and as a way of playing out various pedagogical scenarios as I experimented with student-centered teaching methods that were new to me at the time.

Quite early on many of my students began reading the blog and using it to learn what it is that goes through my mind as I design my courses. I realized then that the blog could serve as a new means of communicating with my students, as a way of making clear my intent and of making my techniques transparent. It could also serve as a means of disseminating my students' work, and of discovering their thoughts on the issues we faced together in the classroom.

For all it's done, writing this blog has been very relaxing and relieving exercise for me, and even if not another soul were to read it, I'd still feel it's worth the time it takes to write it.

But I know I've got some readers out there; I'd just like to know who you are.

So I'd like to ask my regular (or even non-regular) readers to ping me back in the comments section to this post, anonymously if you'd like. Just write me a few words to let me know that you're out there. Feel free to share a bit about yourself, if you'd like to, and tell me what road it is that's brought you here, or what it is that brings you here again. I'm very curious: knowing full well that there are hundreds of far funnier and far more entertaining blogs on the interwebs, I'd like to know what it is that holds your interest enough to make you keep coming back for more.

Where will I go from here? We'll see. I don't intend to change much about the way I put the blog together in the coming months, although I'm toying with the idea of beginning a new blog devoted to poetry, so here you may see a little less of that in the future (unless there's tremendous call to keep it here). I wouldn't have dreamed three years ago that I'd be where I am today in my teaching, so I hesitate to speculate further about what Change of Basis might look like in a year or two. Time will tell.

With that, fair reader, let me end this post. I hope to hear from you, if only in a few words, in the comments section.

Wednesday, August 05, 2009

Technically, I'm ready for Day One...and Day Two, for that matter...in both of my classes, aside from actually printing out the syllabi for Calc I. The first-day worksheets are ready for Foundations, the first-day activities are mapped out for Calc, and both websites are up and running and fully functional.

Nothing to do now but try to get a little bit further ahead in class prep, and maybe fret a bit for the next week and a half.

I suppose there'd be something wrong with me if I didn't worry a little bit.

Sunday, August 02, 2009

This afternoon I put an hour or so into revamping my 280 syllabus for this oh-so-imminent semester.

As I hinted in my last post, I'm going to be asking the students to write their own textbook, in a sense: at the end of every "section" of the class (of which there are seven, listed in the syllabus excerpt below) I'll ask for several students to serve as the primary authors of a "chapter" of the "textbook" that will deal with the ideas we've just finished discussing as a class. That chapter will include examples, explanations, exercises, and proofs.

Here's the relevant snippet from the syllabus:

***

I feel that it will be a far more meaningful exercise for you to construct your own textbook for the course, instead of following along in a textbook that's already been written. Therefore, at the end of each "section" (of which there are 7 we'll discuss throughout the course, and they're listed below) I will ask a group of three to five of you to spend roughly a week composing your own definitions, explanatory text, examples, computations, and proofs to create a "chapter" of text. Once you've typeset your chapter using the LaTeX software, you will post it to the course's Moodle [the University's online course management and social networking platform] site, where others will have a chance to offer feedback: do the definitions seem right? Do the examples make sense? Are the proofs correct? Others may ask to permission to edit your chapter (or at least ask you to perform edits). In this sense the textbook you create is a wiki, communally created by you as a class.

Incidentally, although there will be a due date for each initial chapter posting, there will be no due date on revisions once a chapter is posted; therefore once a chapter is up, it's fair game for modification for the rest of the semester.

Here is a list of the "sections" we'll discuss during the semester; you will each be expected to work on two chapters:

Some of these sections are more involved than others; I will assign more of you to collaborate on the more complicated ones, allowing you to more effectively divide the work. However, the writing of each chapter should be a well-coordinated task yielding a well-composed whole; therefore I strongly advise you to select a "chief editor" for each chapter once you're placed on a given team of authors. I'll give you some more tips as the time nears to write the first chapter.

***

I recognize that this addition to the class is going to impose on the students a substantial amount of work, and therefore I'll be looking for ways to streamline the ordinary homework assignments and create meaningful connections between those homeworks and the textbook chapters the students will be asked to write. (For instance, what can I do in writing my own homework problems that will help the students to write their own homework problems for the chapters?) Moreover, I'm thinking of scrapping the end-of-semester presentations in order to help students free up time for "textbook" revisions at the semester's end.

As another time-saver (and means of encouraging student self-authorship), for each worksheet I'll be looking for volunteers to serve as "discussion leaders"; these people will be expected to digest each worksheet as it's handed out and before the class addresses it, and they may be called upon to lead the class in completion of the worksheet's activities in class. This slight change in classroom operations should help in-class activities to run more smoothly.

We'll see how it goes. It should help a lot that this class is by far the smallest of the 280 sections I've taught at UNCA so far: only 14 people are registered right now, and I don't think that's likely to go up.

For the MATH 280 alumni among my readers, what would you think about the above changes? I'd really like to know your thoughts.